This establishes that \(y_h + y_p\) is a solution. Next we need to show that all solutions are of this form. Suppose that \(y_3\) is a solution to the nonhomogeneous differential equation. Then we need to show that

\[ y_3 = y_h + y_p \]

for some constants \(c_1\) and \(c_2\) with

\[ y_h = c_1y_1 + c_2y_2. \]

This is equivalent to

\[ y_3 - y_p = y_h .\]

We have

\[ L(y_3 - y_p) = L(y_3) - L(y_p) = g(t) - g(t) = 0. \]

Therefore \(y_3 - y_p\) is a solution to the homogeneous solution. We can conclude that

\[y_3 - y_p = c_1y_1 + c_2y_2 = y_h.\]

\(\square\)

This theorem provides us with a practical way of finding the general solution to a nonhomogeneous differential equation.

Step 1: Find the general solution \(y_h\) to the homogeneous differential equation.

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